Imagine constructing an open-topped water tank from a square metal sheet (2 m by 2 m). You would cut squares from the four corners of the sheet and bend up the four remaining rectangular pieces to form the sides of the tank. Then you would weld the edges together to make them watertight. Your goal is to construct a tank with the greatest possible volume.

What size squares do you conjecture would result in the water tank with the greatest volume?

To investigate the relationship between the maximum volume of the tank and the size of the squares cut from the corners, build models and collect data. Using a 1:10 scale, start with a model -- 20-by-20-cm square sheets (PDF - be sure to print this document full scale) of paper. Take one sheet of paper and cut a 1-by-1-cm square from each corner. Fold the net into an open box and tape it. You have just constructed a scale model of the water tank. Repeat the process to construct different models of the water tank.

Using whole-number side lengths, which size of a cutout square results in the largest volume for the box? What is the size of the cutout square and the resulting volume?

Problem C3

These models were at a scale of 1:10. If the largest box you made represented the water tank, what would its dimensions be?

Problem C4

One way to get a closer estimate of the dimensions of the box with the greatest volume is to make a graph. On the x-axis, plot the length of the sides of the cutout square (in centimeters); on the y-axis, plot the corresponding volume of the box (in cubic centimeters). Use a graphing calculator, sketch your graph on grid paper (PDF file), or enter the data into the graphing program on your computer.

What would happen if you could remove squares from the corners that used decimals, such as side of square = 3.5 cm, or side = 3.75 cm? Approximate the size of the squares that should be cut to maximize the resulting volume. Note 3

Video SegmentFind out how the course participants went about modeling a container with the maximum volume by cutting out different-sized squares from the 20-by-20 sheet of paper. Jayne and Lori try to use one of their earlier observations about surface area and volume. David Russell and David Cellucci graph the data, which leads to new observations. They consider the effect of the absence of a lid on a relationship between volume and surface area.

Why do you think Jayne and Lori's initial intuitive approach didn't work? How would you explain it in your own words?

If you are using a VCR, you can find this segment on the session video approximately 18 minutes and 3 seconds after the Annenberg Media logo.